Three-dimensional numerical study of the reactive powder concrete segments in tunnel lining

: The tunnel lining systems act as lines of defence against the forces and geotechnical situations. The use of precast concrete tunnel linings (PCTLs) has been escalating due to its effective and economical installation process. The tunnels usually suffer from the premature deterioration due to corrosion of the reinforcement and thus need maintenance. Corrosion leads to the distress in PCTL leading to the cracking and finally the scaling of concrete. This study aims to assess the structural durability performance of reactive powder concrete (RPC) as the material of tunnel lining segments compared to reinforced concrete (RC) and high performance concrete (HPC). The numerical findings indicated that the maximum load capacity of PRC-PCTL segments was greater than that of the corresponding RC and HPC segments. Regarding the findings, PRC is a very significant option for conventional segments. The high strength of PRC can decrease the thickness of the PCTL segments, resulting in the decreased material cost. Also, PRC-PCTL segments can eliminate the laborious and costly production of RC segments and mitigate the corrosion damage and thus enhance the service life of lining segments.


Introduction
The use of precast concrete tunnel segments in the projects has creased due to its effective and economical utilization compared to the conventional in-situ lining appraoch [1][2][3].In general, precast concrete tunnel lining (PCTL) segments are planned for more than 100 years of service life [4] with reinforced concrete (RC).Hence, several cases suffered from the premature deterioration prior to gaining their service life.It was basically assigned to the corrosion caused by chloride infiltration [5][6][7].
The Reactive Powder Concrete (RPC) is the ultra-high strength and ductile concrete prepared by replacing the standard aggregate with quartz powder, silica fume and steel fibers [8].The concrete does not comprise coarse aggregates, but involves cement, silica foam, sand, powdered quartz, superplasticizer, and steel fiber.Avoiding the inclusion of coarse aggregates by the inventors is a significant aspect of the microstructure and function of RPC which decreases the heterogeneity in mortar and aggregate mix.The additives, lack of coarse aggregates, small water-tocement ratio, fine steel fibers, heat treatment, and compression before and after setting are the RPC's fundamental features [9][10][11][12][13][14].The compressive RPC strength is in the 200-800 MPa range, its flexural strength is from 30 to 50 MPa, and its Young's modulus around 50-60 GPa [15][16][17][18][19].Because of RPC's durability and excellent mechani properties, its utilizations are expanded.RPC structural elements are resistant against chemical attack, impact loads (with collision with vehicles and vessels), and sudden kinetic loads based on the earthquakes.Made up of compact and ordered hydrates, RPC is basically characterized by high performance.The microstructure is optimized by accurate particle gradation to maximize density.RPC heavily relies on the pozzolanic features of the washed silica foam and the optimizing cement concrete to gain the maximum hydrate [20][21][22][23][24][25][26][27].Thus, PRC can be proven to be a more longlasting material for precast concrete tunnel lining construction.Besides to enhancing the mechanical and durability features, the replacement of reinforcement with PRC in tunnel segments can delete the laborious and costly construction of tunnel segments.Moreover, PRC lining segments' cross-sectional dimensions can be decreased due to their high strength, leading to economical fabrication.Multiple research investigated the flexural capacity of conventional RC tunnel segments [28][29][30][31][32][33].The corrosion deterioration of conventional RC-PCTL has resulted in developing the reactive powder concrete (PRC) tunnel lining segments.Hence, the present research obtains an evaluation of the mechanical behaviour of reactive powder concrete (RPC) tunnel lining segments.This study aims to assess the structural and durability function of reactive powder concrete (RPC) tunnel lining segments compared to reinforced concrete (RC) and high performance concrete (HPC).Furthermore, finite element analysis using ABAQUS was conducted in order to verify the experimental behaviour and comparison of the performance of RPC with RC and HPC tunnel lining segments was presented in this study.

RPC in comparison with HPC
A comparison between mechanical and durability properties of the RPC and (high performance concrete) HPC shows the RPC to offer a higher compressive strength and a permeability lower than the HPC.The HPC is a product of novel concrete science, which uses additives and scientific techniques to control the concrete's microstructure [34,35].
Owing to its microstructure, the HPC has achieved its maximum compressive strength.Nonetheless, at any given strength, coarse aggregates are the weakest links in the concrete structure.The only remaining solution to further increase the compression strength of the concrete is to remove coarse aggregates, which is the same approach that has been adopted in developing the RPC.Table 1 compares the mechanical properties of the typical RPC with a regular HPC with an 80 MPa compressive strength.The higher fracture toughn of the RPC is suggestive of its better ductility.Aside from the mechanical properties, RPCs have an exceptionally dense microstructure that makes them impermeable to water and durable.
Thanks to its low porosity and permeability, restricted contraction, and high corrosion resistance, the RPC offers remarkable durability.Compared to the HPC, the RPC passes no liquid or gas through.The RPC specifications listed under Table 2 enable it to be used in aggressive chemical environments and applications where other types of concrete do not last long due to wear and tear.Results indicate that RC and HPC concretes are more vulnerable to corrosion problem compared with RPC concrete.

Numerical verification
The finite-element software, ABAQUS [36] was applied for model verification, contrasting the findings with the experimental surveys of Abbas [37].

Model geometry
Figure 1 illustrates the model geometry, and the meshed model is presented in Figure 2. The model was meshed using square elements with a length-to-width ratio of 1, as it has been suggested that they produce the best results

Model boundary conditions
As depicted in Figure 3, support conditions were defined for the model by fixing the model in the vertical direction (Figure 4) at the points where the segment is in contact with two bottom frames (reaction).Further, a vertical downward velocity was defined for the upper part of the segment (Figure 5) for flexural loading.A critical parameter in the finite-element techniques is the solution time and complexity, which depend on the shape and number of elements.In light of the above discussions, different types of mesh were used in modeling the studied tunnel segment.It was found that similarly acceptable results are obtained by using 50 and 25 mm elements.Therefore, the 50 mm elements were used for the shorter analysis time (Figure 6).

Result validation
The current numerical ABAQUS [36] model (Figure 6) was contrasted with the load-displacement curve of the flexural test in the work of Abbas [37] (Figure 7) at the point of loading for validation.The two curves are plotted on the same graph in Figure 8 for a better comparison.Comparing two curves indicate the consistency of this study's findings with those of Abbas [37].
Three-dimensional numerical study of the reactive powder concrete segments in tunnel lining | 289

Optimizing the tunnel segments with RPC
In this section, the impact of incorporating the RPC in the tunnel segments are studied with 3D modeling.To this objective, a 3D model of subway tunnel with actual dimension was developed applying ABAQUS [36] Figure 9 indicates the segment model with the details about the overhead, tunnel diameter, and lining thickness.Table 3 shows the details of developed models.

Model geometry and meshing
Figure 9 illustrates the model geometry, and the meshed model is presented in Figure 10.The model comprises six layers of three different materials.The model was meshed considering the two following points: First, square elements (with a length-to-width ratio of 1) were used for meshing, as they have been recommended for producing the best results in the finite-element method.Second, a finer mesh was used for the tunnel due to the considerable displacement and stress concentration (Figure 11).On the other hand, a smaller element with finer mesh will be closer to representing distributed loading type, as the nodes will be closer to the locations of the contact.In addition, as a general principle of any numerical method, the finer the discretization the closer it is to the exact solution.

Material and element specifications
Table 4 lists the specifications of the materials used to model soil layers.As evident, elastic modulus, Poisson's ratio, cohesion, and internal friction angle are necessary to model the soil.The linearly elastic-perfectly plastic model using the Coulomb constitutive law was used to represent the soil.Further, the concrete was modeled using the linear elastic relationship based on Table 5.In this case, the elastic modulus, Poisson's ratio, and the specimens' compressive strength are necessary.Interactions between the soil and the concrete cover were modeled as follows: Tangential behavior: The tangential behavior between the soil and the concrete followed Coulomb's friction law in zero-cohesion conditions with a µ = 100 (ϕ = 89.43∘ ) coefficient of friction (Figure 12a).This relationship has been recommended by the ABAQUS [36].
Vertical Behavior: The hard contact relation was used to model the vertical behavior, as evident from the figure below.This configuration allows for the vertical transfer of stress in case of contact between two surfaces.The model does not account for tensile stress (Figure 12b).

Model boundary conditions
The support conditions were defined to fix the displacement and rotation of the model's bottom in the horizontal and vertical directions.Further, the sides were fixed in the horizontal direction.Figure 13 shows the defined boundary conditions.The boundaries have been placed far enough from the tunnel to ensure that the boundaries have negligible effects in modeling.

Initial conditions of the model
The initial conditions of the problem include in-situ stresses and groundwater conditions.According to Figure 14, the model assumes a 19.65 m water table.The model's pore water pressure was based on the water level.In the first stage, static analysis, in-situ stresses were created in the model.Hence, RPC withstands 48, 38, and 20% higher axial force than RC, HPC, and UHPC, respectively.Thus, considering the RPC's excellent performance, a smaller segment thickness can be regarded in the design than with other kinds of concrete.

Shear force on the tunnel lining
The shear force on the tunnel lining for multiple segments is plotted in Figure 16.The amounts in the figure are in kN.The maximum shear in the models corresponds to the case with the RPC and is about 30% higher than other kinds of concrete.Almost the same shear force is exerted on the tunnel lining in the other cases, and similarly to the argument made about the axial force, the higher shear capacity aids make lighter tunnel segments.

Bending moment on the tunnel lining
The bending moment on the tunnel lining for different segments is plotted in Figure 17.The values in the figure are in kN•m.The findings indicate that the maximum axial force corresponds to the RPC and the minimum axial force to the regular concrete.A same argument to that created for the shear stress on the tunnel lining holds for the bending moment, as the higher control over the displacements and the deformation around the tunnel increases the maximum bending moment in the case of the reinforced concrete.Moreover, RPC withstands 46, 48, and 26% higher axial force than regular concrete, HPC, and UHPC, respectively.Thus, given the RPC's excellent performance, a smaller segment thickness can be regarded in the design than with other kinds of concrete.

Displacement on the tunnel lining
The displacement on the tunnel lining in multiple cases is plotted in Figure 18.Considering the displacement symmetry on the tunnel lining, the figure plots the displacement for only half of the tunnel segment.The values in the figures are in mm.It is clear from Figure 18 that horizontal displacement of tunnel lining is decreased by reinforcing the tunnel segment with RPC.

Conclusions
The impacts of applying the RPC in a tunnel segment were surveyed by 3D modeling utilizing the finite-element soft-ware ABAQUS.The numerical models were approved via experimental results.Results show that reinforced concrete-PCTL segments are more vulnerable to corrosion problem compared with RPC-PCTL.The numerical findings indicated that the compressive strength of PRC segments was greater than that of the RC and HPC segments.Regarding the findings, PRC is a very significant option for conventional RC-PCTL segments.Very high strength of PRC can permit decreasing the thickness of PCTL segments, resulting in the decreased material cost and more sustainable fabrication.Besides, PRC-PCTL segments can delete the laborious and costly manufacturing of RC segments which mitigates the corrosion damage, resulting in the improved service life of tunnel segments.The findings indicate that the maximum axial force corresponds to the RPC and the minimum axial force to the conventional RC.The same argument to which created for the shear stress on the tunnel lining holds for the bending moment, as the higher control over the displacements and the deformation around the tunnel increases the maximum bending moment in the case of the reinforced concrete.Moreover, the RPC withstands 48, 38, and 20% higher axial force than RC, HPC, and UHPC, respectively.Hence, considering the RPC's excellent performance, a smaller segment thickness can be regarded in the design than with other kinds of concrete.

Figure 3 :Figure 4 :
Figure 3: Experimental boundary conditions in the lower part of the segment Abbas[37]

Figure 5 :
Figure 5: Numerical boundary conditions in the upper part of the segment

Figure 6 :
Figure 6: Load-displacement curve of the model with two mesh sizes

Figure 8 : 3 :
Figure 8: A comparison of the load-displacement curves

Figure 12 :
Figure 12: Coulomb friction law and vertical behavior between the soil and the concrete (hard contact) ABAQUS (2012)

Figure 13 :Figure 14 :
Figure 13: Boundary conditions on the sides

Figure 17 :
Figure 17: Comparing the bending moment in different models

Figure 18 :
Figure 18: Horizontal displacement on the tunnel lining in different models

Table 1 :
Mechanical properties of RPC vs. HPC

Table 2 :
The durability of RPC vs. HPC

Table 4 :
Soil layer specifications

Table 5 :
Concrete specifications in the models